why does series (-1)^n+1 from n =1 to infinity diverge?

The Cesaro sum is 1/2

http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation

For an arbitrarily large n your series may sum to 1 or to 0. And for n+1 to 0 or to 1. That isn't converging...

conventionally for a series to converge lim n--> infinity |Sn| = 0

You have a nice alternating series. Now, there's a theorem which says that the series of some expression converges only if the series of the absolute value of the expression converges.

In this case, -1^(n+1) = 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ...

Consider |-1^(n+1| = 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... As you can see, the series associated with this sequence one diverges.

Therefore, the original expression diverges.

You can also see that the original series is not Cauchy (there is no moment in the sequence in which the distance of an element with all the others after it is less or equal to 1).

In order to converge to a limit, you need to get closer and closer to that limit, and that means that the terms have to get closer and closer together.

This series oscillates between two values. It never gets any closer to anything.

There aren't really "steps", there's merely a one-line argument that it doesn't meet the definition of "convergence".

Because its nth term does not even converge to zero, so it cannot possibly converge.

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